Conceived of philosophically, the foundations of mathematics concern various metaphysical and epistemological problems raised by mathematical practice, its results and applications. Most of these problems are of ancient ...
Nonstandard analysis is an important application of mathematical logic to the rest of mathematics. Invented in 1960, it provided a long-sought-for rigorous justification for the use of infinitely ...
The term ‘mathematical analysis’ refers to the major branch of mathematics which is concerned with the theory of functions and includes the differential and integral calculus. Analysis and ...
Realism in the philosophy of mathematics is the position that takes mathematics at face value. According to realists, mathematics is the science of mathematical objects (numbers, sets, lines ...
The philosophy of arithmetic gains its special character from issues arising out of the status of the principle of mathematical induction. Indeed, it is just at the point ...
The axiom of choice is a mathematical postulate about sets: for each family of non-empty sets, there exists a function selecting one member from each set in the ...
Beth’s theorem is a central result about definability of non-logical symbols in classical first-order theories. It states that a symbol P is implicitly defined by a theory T ...
Boolean algebra, or the algebra of logic, was devised by the English mathematician George Boole (1815–64) and embodies the first successful application of algebraic methods to logic. ...
Buddhist philosophers have investigated the techniques and methodologies of debate and argumentation which are important aspects of Buddhist intellectual life. This was particularly the case in India, where ...
Cantor’s theorem states that the cardinal number (‘size’) of the set of subsets of any set is greater than the cardinal number of the set itself. So once ...
Since the 1960s Lawvere has distinguished two senses of the foundations of mathematics. Logical foundations use formal axioms to organize the subject. The other sense aims to survey ...
A ‘category’, in the mathematical sense, is a universe of structures and transformations. Category theory treats such a universe simply in terms of the network of transformations. For ...
Chaos theory is the name given to the scientific investigation of mathematically simple systems that exhibit complex and unpredictable behaviour. Since the 1970s these systems have been used ...
Church’s theorem, published in 1936, states that the set of valid formulas of first-order logic is not effectively decidable: there is no method or algorithm for deciding which ...
An algorithm or mechanical procedure A is said to ‘compute’ a function f if, for any n in the domain of f, when given n as input, A ...
Combinatory logic comprises a battery of formalisms for expressing and studying properties of operations constitutive to contemporary logic and its applications. The sole syntactic category in combinatory logic ...
The theory of computational complexity is concerned with estimating the resources a computer needs to solve a given problem. The basic resources are time (number of steps executed) ...
The standard definition of randomness as considered in probability theory and used, for example, in quantum mechanics, allows one to speak of a process (such as tossing a ...
The effective calculability of number-theoretic functions such as addition and multiplication has always been recognized, and for that judgment a rigorous notion of ‘computable function’ is not required. ...
At first sight, computers would seem to be of minimal philosophical importance; mere symbol manipulators that do the sort of things that we can do anyway, only faster ...
The idea of one proposition’s following from others – of their implying it – is central to argument. It is, however, an idea that comes with a history ...
Constructivism is not a matter of principles: there are no specifically constructive mathematical axioms which all constructivists accept. Even so, it is traditional to view constructivists as insisting, ...
The ‘continuum hypothesis’ (CH) asserts that there is no set intermediate in cardinality (‘size’) between the set of real numbers (the ‘continuum’) and the set of natural numbers. ...
How is it known that every number has a successor, that straight lines can intersect each other no more than once, that causes precede their events, and that ...
Decision theory studies individual decision-making in situations in which an individual’s choice neither affects nor is affected by other individuals’ choices; while game theory studies decision-making in situations ...